Semigroup Theory Of Linear Operators Research Articles

Parabolic PDEs on low-dimensional structures

The paper concerns the theory of parabolic equations on a broad class of closed subsets of Euclidean space possessing a kind of tangent structure. A necessary framework for considering evolutionary problems is developed, and fundamental results about the existence and uniqueness of solutions are established. The presented approach is based on applying the semigroup theory of linear operators combined with special geometric constructions related to the shape of the examined structure. The proposed setting is consistent with the theory of elliptic equations on lower dimensional structures and with the second-order calculus of variations developed by Bouchitté and Fragalà in [5].

Asymptotic behavior and numerical analysis for a Timoshenko beam with viscoelasticity and thermodiffusion effects

We study the qualitative and numerical behavior of a thermodiffusion Timoshenko beam with Kelvin–Voigt damping. The model, recently derived by Aouadi $ \rm et al. $ (2019), describes the behavior of a Timoshenko beam with thermodiffusion effects under the classical Fourier's and Fick's laws for heat and mass diffusion transmission, respectively. With the help of the semigroup theory of linear operators, we prove the well posedness of the considered problem. Without assuming the well-known condition of equal wave speeds, it is shown that the associated $ C_0 $-semigroup to the fully viscoelastic system is analytic. Exponential stability and impossibility of localization of the solutions in time are immediate consequences. Two partial viscoelastic systems are shown to be polynomially or exponentially stable. The approach based on the Gearhart–Herbst–Prüss–Huang theorem is used to show the lack of exponential stability, whereas the optimal polynomial decay rate can be shown using the Borichev & Tomilov's Theorem. Finally, we propose a spectral discretization method to prove that the associated discrete energy decays exponentially or polynomially otherwise.

Asymptotic behavior of the Rao–Nakra sandwich beam model with Kelvin–Voigt damping

The Rao–Nakra sandwich beam is a coupled system consisting of two wave equations for the longitudinal displacements of the top and bottom layers and an Euler–Bernoulli beam equation for the transversal displacement. This paper concerns the system’s stability when the Kelvin–Voigt damping terms act on the first and third equations. Using the semigroup theory of linear operators, we prove the global well-posedness of the associated initial boundary value problem. And then, we prove the lack of exponential stability of the system. Because of this lack of exponential stability, we study the polynomial stability and prove that the system decays with rate . We further prove that this decay rate is optimal.

Boundary control of an axially moving hybrid system of elasticity with mass-rotary rollers

In this paper, we investigate various stabilisation problems of a hybrid system of elasticity, consisting of a thin tape, which moves axially between two sets of rollers. We present a control solution, which works simultaneously, for three variants of the tape model. In particular, we design boundary feedback control laws via Lyapunov's second method. The resulting closed-loop system for the general case is then analysed, where we prove the existence and uniqueness of solution using the semigroup theory of linear operators; furthermore, we prove the exponential decay of the solution via a frequency domain theorem. The stabilisation of the two other special cases is derived implicitly from the general case proofs presented in the paper. Finally, the paper is concluded by presenting computer simulations to support the theoretical results.

Exponential stability for a nonlinear porous-elastic system with delay

In this work, we consider the existence of global solution and the exponential decay of a nonlinear porous elastic system with time delay. The nonlinear term as well as the delay acting in the equation of the volume fraction. In order to obtain the existence and uniqueness of a global solution, we will use the semigroup theory of linear operators and under a certain relation involving the coefficients of the system together with a Lyapunov functional, we will establish the exponential decay of the energy associated to the system.

Existence and Stability Results of Stochastic Differential Equations with Non-instantaneous Impulse and Poisson jumps

Abstract This paper focuses on a new class of non-instantaneous impulsive stochastic differential equations generated by mixed fractional Brownian motion with poisson jump in real separable Hilbert space. A set of sufficient conditions are generated based on the stochastic analysis technique, analytic semigroup theory of linear operators, fractional power of operators, and fixed point theory to obtain existence and uniqueness results of mild solutions for the considered system. Furthermore, the asymptotic behaviour of the system is investigated. Finally, an example is proposed to validate the obtained results.

Analyticity of solutions to thermo‐elastic‐plastic flow problem with microtemperatures

AbstractIn this paper, we study some qualitative and numerical properties of new equations including the coupled effects of thermal elastic‐plastic theory with microtemperatures. We establish the necessary and sufficient conditions to guarantee that the model dissipates energy. The one‐dimensional case, which corresponds to isotropic hardening problem, is chosen in order to present some qualitative and numerical properties. With the help of the semigroup theory of linear operators, we prove the well‐posedness of the one‐dimensional problem corresponding to plastic flow. Then, we show that the associated semigroup is not analytical in general, except for a special case. The exponential stability of the solutions is kept in all cases. Finally, a numerical tool, based on the finite element method, is developed to validate the proposed model and to show its capability. Particular attention is paid to the consideration of the elastoplastic behavior in the development of this tool.

Existence, stability and controllability results of stochastic differential equations with non-instantaneous impulses

ABSTRACT This paper is devoted to the study of a new class of non-instantaneous impulsive stochastic differential equations driven by mixed fractional Brownian motion in separable Hilbert spaces. Based on the stochastic analysis theory, analytic semigroup theory of linear operators, fractional powers of operators, and a fixed point technique, a new set of sufficient conditions are derived to ensure the existence and uniqueness of mild solutions for the proposed stochastic system. Moreover, we also investigate the asymptotic behaviour of mild solutions and controllability results for the proposed stochastic system. Finally, an example is given to demonstrate the applicability of our main results.

Asymptotic Stability of Singular Solution for Camassa-Holm Equation

The aim of this paper is to study singular dynamics of solutions of Camassa-Holm equation. Based on the semigroup theory of linear operators and Banach contraction mapping principle, we prove the asymptotic stability of the explicit singular solution of Camassa-Holm equation.

Inverse Semi-group of Normal Sub-semigroup Based on Computer

Semi-group theory of linear operators is the theoretical basis and an important tool for the research of repairable systems, which plays an essential role especially in discussing the suitability, asymptotics, and stability of system solutions. In this paper, a class of repairable computer system models is studied based on the normal semi-group theory. The theory of normal sub-semigroup is used to discuss the inverse semi-group features of system operators. The results show that system operators have closedness, consistency, and dissipativity.

Dynamic analysis of a complex system under preemptive repeat repair discipline

This paper considers a two-unit complex system, in which one of the components has priority with preemptive repeat repair disciplines, described by partial differential equations with integral boundary conditions. First, we prove that the system has a unique nonnegative time-dependent solution by using the strong continuous semigroup theory of linear operators. Then, we obtain the exponential convergence of the time-dependent solution to its steady-state solution by means of the spectral properties of the corresponding operator. We also provide the asymptotic behavior of some time-dependent reliability indices, a numerical illustration showing the effects of various parameters on the system, and the validity of the theoretical analysis.

Boundary Stabilization of a Thermoelastic Diffusion System of Type II

In this paper we study the boundary stabilization of a one-dimensional thermoelastic diffusion problem of type II. The system of equations is a coupling of three hyperbolic equations. This poses some new mathematical and numerical difficulties. With the help of the semigroup theory of linear operators, we prove the well-posedness of the proposed problem. By using the frequency domain method combined with the multiplier technique, we prove the exponential stability of the solutions. Finally, we present a numerical scheme based on the Chebyshev spectral method and we give two numerical examples to validate the proposed model and to show its capability.

Analyticity of solutions to thermoviscoelastic diffusion mixtures problem in higher dimension

In this paper, we consider the linear theory of binary mixtures for thermoviscoelastic diffusion materials derived by Aouadi et al. (J Therm Stress 41:1414–1431, 2018). We establish the necessary and sufficient conditions to get a dissipation inequality for isotropic centrosymmetric materials. With the help of the semigroup theory of linear operators, we prove the well posedness of the higher-dimensional problem. Then, we show that the associated $$C_0$$-semigroup is analytic. Exponential stability and impossibility of localization of the solutions in time are immediate consequences.

Analytical aspects in strain gradient theory for chiral Cosserat thermoelastic materials within three Green-Naghdi models

This work is motivated by the recent interest in using strain gradient theory to model the chiral behavior of elastic materials. In this paper, we derive a linear strain gradient theory for Cosserat thermoelastic materials according to the three models (types I, II and III) of Green-Naghdi theory. Models II and III permit propagation of thermal waves at finite speeds, while model I coincides with the classical Fourier’s law. The thermal field is influenced by the displacement and the microrotation fields and by some additional parameters that describe the chiral behavior. We prove the well-posedness for the three models and the asymptotic behavior for models I and III by the semigroup theory of linear operators.

Well‐posedness and energy decay of solutions to a wave equation with a general boundary control of diffusive type

AbstractIn this paper, we study well‐posedness and asymptotic stability of a wave equation with a general boundary control condition of diffusive type. We prove that the system lacks exponential stability. Furthermore, we show an explicit and general decay rate result, using the semigroup theory of linear operators and an estimate on the resolvent of the generator associated with the semigroup.

Viscoelastic materials with a double porosity structure

This paper in concerned with the linear theory of materials with memory that possess a double porosity structure. First, the formulation of the initial-boundary-value problem is presented. Then, a uniqueness result is established. The semigroup theory of linear operators is used to prove existence and continuous dependence of solutions. A minimum principle for the dynamical theory is also derived.

Asymptotic behavior of repairable device systems under warranty and beyond warranty periods

In this paper, we present a new mathematical model generalizing the model of a single unit repairable device system, with preventive maintenance under warranty period, and the model of a repairable multi-state device system. To study the time dependance solution of the resulting system, we present in this paper a new analytic approach. So, by using the $$C_0$$ -semigroup theory of linear operators and some spectral properties, we proof that the dynamic solution of the system converges to steady state solution when the time go to infinity.

Well-posedness and exponential stability in binary mixtures theory for thermoviscoelastic diffusion materials

A nonlinear theory is derived for a thermoviscoelastic diffusion composite which is modeled as a binary mixture consisting of two Kelvin–Voigt viscoelastic materials. First, the basic equations of the nonlinear theory of heat and diffusion conducting viscoelastic mixtures are derived in Lagrangian description. Then, the theory is linearized and field equations are given for both anisotropic and isotropic centrosymmetric materials. We establish the necessary and sufficient conditions to get a dissipation inequality for isotropic centrosymmetric materials. With the help of the semigroup theory of linear operators, we establish the suitable framework where the linear problem is well posed. Finally, we prove that generically the solution decreases exponentially thanks to the viscous dissipation effects of the constituents.

Long-time dynamics for nonlinear porous thermoelasticity with second sound and delay

This paper is concerned with the long-time behavior of a damped porous thermoelastic problem. It has been studied by many authors and most of the known results are concerned with decay issues under different boundary conditions and damping terms. There are just a few references on the long-time dynamics of such systems. The purpose of the present work is to complement and extend some early studies on porous thermoelastic systems by establishing new results on the existence of attractors and some of their properties. Motivated by this scenario, we consider a one-dimensional porous thermoelastic system with linear frictional damping, nonlinear source terms, and a time-varying delay term in the internal feedback, where the heat flux depends on Cattaneo’ law. Under some suitable assumptions on the weights of feedback, we establish the global well-posedness of the system by using the C0−semigroup theory of linear operators. Then we show the existence of a global attractor for finite energy solutions and we prove its smoothness and finite fractal dimension. Furthermore, the existence of a generalized fractal exponential attractor is also derived.

Stability aspects in strain gradient theory of thermoelasticity with mass diffusion

AbstractThis article is concerned with a strain gradient theory for thermoelastic diffusion materials. The work is motivated by the recent interest in the study of gradient theories and increasing use of materials which possess thermal and mass diffusion variations. First, we establish the basic equations of the nonlinear strain gradient theory for thermoelastic diffusion materials. Then, we deduce the constitutive equations for isotropic chiral thermoelastic diffusion materials. With the help of the semigroup theory of linear operators, we prove the well‐posedness of the problem and the asymptotic behavior of the solutions. The exponential stability is proved for the one‐dimensional problem by a spectral method.